16 problems

Let $F$ be an infinite field, let $k$ be either $\mathbb{Q}$ or $\mathbb{Z}/l$ for a prime $l$, and assume $K_2^M(F)\otimes k=0$. Let $\mathfrak{t}_r^{(k)}(F)$ denote the homologic…

Kato–Kuzumaki's conjecture. A field $k$ satisfies $C_i^q$ if and only if

Let $A$ be a local ring, let $i\geq 0$ be an integer, let $\widehat{\text{K}}{}^{\text{M}}_i(A)$ denote the improved Milnor $K$-group, and let…

Let $k$ be a flexible field and let $F/k$ be a finitely generated field extension. The isotropic Milnor K-ring of $F/k$, denoted $\operatorname{\mathrm{k}}^M_*(F/k)$, is the quotie…

Let $q$ be a non-negative integer and let $K$ be a field. Say that $K$ has the $C^q_{\mathrm{sc}}$ property if, for every finite extension $L/K$ and every principal homogeneous spa…

Graph isomorphism conjecture. For every such $A$, the graph homomorphism is an isomorphism.

Let $A$ be a henselian discrete valuation ring of characteristic zero with residue field of characteristic $p$ and function field $K$. Let $X$ be smooth and projective of relative…

Let $A$ be an excellent henselian discrete valuation ring with uniformising parameter $\pi$ and residue field $k$, let $X$ be a smooth projective scheme over…

Bloch–Quillen formula. The cycle class map $\operatorname{cyc}_X^M$ is an isomorphism. Together with the known surjectivity, this would identify the Levine–Weibel Chow group of zer…

Weak Lefschetz conjecture. This map is an isomorphism when $2p<\dim(L)$ and is injective modulo torsion when $2p=\dim(L)$. This is the Chow-group analogue of the classical Weak Lef…

Let $\mathbf{k}$ be a field of characteristic $0$. For a field extension $F/\mathbf{k}$, write … A dominant rational map induces pullback on Milnor $K$-groups. The logarithmic Miln…

Let $K$ be a field and let $l$ be prime such that $K$ contains a primitive $l$-th root of unity. A positively graded algebra $A=k\oplus A_1\oplus A_2\oplus\dotsb$ is Koszul when…

Let $X$ be a projective complex scheme of dimension $d$. Let $F^dKH^M_0(X)$ be the degree-$d$ part of the Brown filtration on Milnor $KH$-theory, and let … be the Chern-class map.…

Let $X$ be a quasi-projective scheme of dimension $d$ over an algebraically closed field $k$ of characteristic zero. Let $CH^d(X)$ denote the Chow group of zero-cycles and let…

Bloch–Kato conjecture. The map $h_{k,n}^i$ is bijective. This conjecture identifies Milnor $K$-theory modulo $n$ with the corresponding Galois cohomology group; its resolution is a…

Bloch–Kato conjecture. This norm residue homomorphism is an isomorphism for any nonnegative integer $n$.